Problem 44 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$y=2+m(x+3) \quad \text { for } m=0, \pm 0.5, \pm 1, \pm 2, \pm 6$$

Step-by-Step Solution

Answer

All lines intersect at the point \((-3, 2)\).

Step 1: Understand the Equation

The equation given is in the form \(y = 2 + m(x + 3)\). This can be rearranged to \(y = mx + (3m + 2)\), which represents a line with slope \(m\) and y-intercept \(3m + 2\). The parameter \(m\) changes the slope, creating different lines.

Step 2: Set up Different Lines

Substitute each value of \(m\) into the equation \(y = 2 + m(x + 3)\) to create specific equations:- If \(m = 0\): \(y = 2\)- If \(m = 0.5\): \(y = 0.5x + 3.5\)- If \(m = -0.5\): \(y = -0.5x + 0.5\)- If \(m = 1\): \(y = x + 5\)- If \(m = -1\): \(y = -x - 1\)- If \(m = 2\): \(y = 2x + 8\)- If \(m = -2\): \(y = -2x - 4\)- If \(m = 6\): \(y = 6x + 20\)- If \(m = -6\): \(y = -6x - 16\)

Step 3: Graph the Lines

Using a graphing device, plot each line in the same viewing rectangle. Make sure to use a consistent scale for both the x-axis and the y-axis, so that the lines can be compared visually.

Step 4: Analyze the Graph

Upon graphing, you will notice that all the lines intersect the point \((-3, 2)\) because if you substitute \(x = -3\) into \(y = 2 + m(x + 3)\), the expression simplifies to \(y = 2\) for any value of \(m\). Thus, \((-3, 2)\) is a common point of intersection for all lines in the family.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an invaluable framework used to express straight lines. It’s presented as \(y = mx + b\), where:

• \(m\) represents the slope of the line, indicating its steepness and direction.
• \(b\) denotes the y-intercept, where the line crosses the y-axis.

In the exercise, the equation \(y = 2 + m(x + 3)\) is rearranged to look more like the slope-intercept form: \(y = mx + (3m + 2)\). Here, only the slope \(m\) changes, but the linear equation still maintains its form, making it easy to understand how the line behaves. This form helps identify both the steepness and where the line meets the y-axis, making graphing efficient.

Graphing Calculator

A graphing calculator is a powerful tool that assists in visualizing mathematical functions, including linear equations. When you input a function, it plots the line on a coordinate plane, showing you the shape and intersection points of the line visually.
For the exercise, using a graphing calculator allows for:

• Quickly plotting multiple lines on the same axis to compare differences.
• Adjusting values like slope dynamically to see how changes affect the line.
• Focusing on specific intersections with tools to zoom in on certain areas.

Using a graphing tool not only saves time but also helps in understanding how changes in the equation parameters affect the graph, providing immediate visual feedback.

Point of Intersection

In the context of linear equations, a point of intersection is where two or more lines meet on a graph. It’s a crucial concept when analyzing families of lines, as these intersections can signify common algebraic properties.
For the given family of lines \(y = 2 + m(x + 3)\), regardless of the value of \(m\), all lines pass through the point \((-3, 2)\). By substituting \(x = -3\) in the equation, the equation simplifies to \(y = 2\) for any \(m\), showing that this point is indeed shared by every line in this set. This common intersection helps illustrate that while slope \(m\) changes, specific coordinate points remain constant, showcasing the role of parameters in defining line behavior.

Family of Linear Equations

When dealing with a family of linear equations, we refer to a set of lines that share a specific relationship but differ based on certain parameters, such as slope or intercept. This concept becomes evident when equations are in a similar form but possess variable elements.
For the given exercise, the family of lines is expressed by \(y = 2 + m(x + 3)\), where the only changing factor is the slope \(m\). Despite these variations:

• Each line retains a linear form, providing consistency across the family.
• The shared intersection point \((-3, 2)\) highlights a uniform characteristic.

This family of lines exemplifies how a single parameter can define a multitude of similar lines, offering insight into broader mathematical patterns and behaviors.